Second, the limit of \(\tau \rightarrow \infty \)  as the lower bound of integration in (REF ) is the signature Markov approximation. Whether such a replacement is reasonable, though, requires justification. The results of our one-dimensional model at oceanic amplitudes suggest otherwise, but this requires defining the relevant time scale for the sake of comparison. We believe this time scale to be \(m/\langle \dot{m} \rangle \)  since the wave frequency enters into the lag-autocorrelation time scale (REF ) and vertical wavenumber and frequency are related through the dispersion relation. This is a by-product of the ad hoc construction of a spatial scale separation (REF ) criterion in our one-dimensional model to supplant a dynamically self consistent specification of the wave packet envelope. On the other hand, if downscale transports are reduced by an order of magnitude, which is entirely reasonable considering the disparity between the prediction (REF ) and community wisdom articulated in [1]}, then an extended one-dimensional closure could fit within the domain of an EDQNM scheme.
